Talk:Rubik's Cube group

Major Revision

I've been extensively revising the Rubik's Cube group page for the last month or so. I think I'm more or less done with the first 3 sections and I'm getting ready to attack the 4th section.

Basically, if the Rubik's Cube is going to be mentioned by the Group article, a Featured Article, it should be a good understandable example. I'm aiming for WP:GA status. When the 4th section is understandable this article should be there.

Any help/reviews are appreciated.--Olsonist (talk) 23:03, 5 April 2012 (UTC)

Confusing

This article is extremely confusing. It throws around such terms as "legal", "fixed", and "orientation" without explaining or even defining them. Melchoir 03:48, 13 May 2006 (UTC)

First sentence

I do not understand the first sentence

In mathematics, the Rubik's Cube is an interesting object because it provides a tangible representation of a mathematical group.

Should'nt one say

In mathematics, the Rubik's Cube group is an interesting object because it both

• gives insight into the mathematics behind the popular Rubik's Cube mechanical puzzle and
• provides a tangible representation of a mathematical group.

--193.175.8.13 16:36, 26 October 2006 (UTC)

"Formal Description" section (don't worry about this, I was wrong)

Center pieces should factor into the number of permutations because they do indeed move in 90 degree orientations. This may not be significant on normally colored cubes, however, not all cubes are normally colored. For instance, this becomes significant on a cube printed with a picture. —Preceding unsigned comment added by 12.19.153.33 (talk) 13:11, 3 October 2008 (UTC)

I can't help thinking that the group G, defined as it is here, is not very useful for understanding the cube, as it neglects the orientation of facets, i.e. flipping a corner (by composing rotations) gives you the same element of G as doing nothing. In short, the identity element of G does not necessarily correspond to the solved state of the cube. The "structure" section below does not make this mistake, but I do not yet know enough group theory to understand that section.

-- by Yeti on23:20 GMT, 04 December 2008 —Preceding unsigned comment added by 137.205.17.152 (talk) 23:21, 4 December 2007 (UTC)

What do you mean by "flipping" a corner? Rotating a corner cube certainly does give a different element of the group, as the three facets on that corner cube will be interchanged. (Well, in fact rotation of a single corner cube is not in G at all; but a matching rotation of two corners is). –Henning Makholm 13:19, 5 December 2007 (UTC)

--: On second thoughts, you are quite right, It does indeed yield a different element, sorry for confusing you all (by Yeti).

One-to-one correspondence

However, there is a one-to-one correspondence between elements of the cube group and positions of the Rubik's cube. Any element of the cube group is a permutation that when applied to the solved cube results in a (legal) cube position. Conversely, any legal cube position must be the result of some sequence of face rotations applied to the solved cube, and any such sequence is an element of the cube group.

I do not think the second part of this statement is quite clear enough. I had too look at it a few times before realizing that it equated each permutation which took something to the same place (i.e. making a 1/4 turn in one direction is the same as making a 3/4 turn in the other). Maybe I was just being slow for not realizing that right away, but I still think it should be made a little clearer, since otherwise the injectivity of such a map is not obvious (or true, for that matter, since there are many more permutations in that case than there are solutions). Cheers. Crito2161 (talk) 03:39, 28 March 2008 (UTC)

Function composition?

I know nothing about group theory, so I hope my wikilinking of "composition" onto "function composition" was correct. Please check. --Slashme (talk) 09:15, 29 July 2008 (UTC)

No, it s composition in the sense of algebra operations. 212.68.238.139 (talk) 11:33, 19 February 2012 (UTC)

For subgroups of the symmetric group, the multaplicative operation of composition is exactly composition of functions. — Preceding unsigned comment added by 160.39.158.105 (talk) 04:50, 1 May 2012 (UTC)

I fixed some, but it still needs a lot of work

I axed the section involving the group axioms -- it has no business being here, and it was very mathematically poor anyway. I merged the rest of it into the section on the group structure, while fixing up some of the math. Still, this is overall a fairly poorly written article and could use a lot more work.— Preceding unsigned comment added by 109.130.66.144 (talk • contribs) 16:49, 27 June 2013‎

Smallest generating set

Does anyone know of a source which gives the size of the smallest generating set of this group? If so, I think that would be an interesting piece of information to add to the article. —Granger (talk · contribs) 20:09, 26 December 2014 (UTC)

I've seen someone claim a set with 2 elements on reddit. I don't remember where exactly though. Judith Sunrise (talk) 23:10, 11 October 2018 (UTC)

Fixed orientation

@Mr. Granger:. I tend to agree with your request for clarification on "moves which can change the positions of the blocks, but leave the orientation fixed". A group such as,

$FDF^{-1}UFD^{-1}F{-1}U^{2}FDF^{-1}UFD^{-1}F^{-1}=(\mathrm {URF,UBR,UFL} )$

has an obvious natural orientation, all the U faces remain in the U orientation. But what about,

$FD^{-1}F^{-1}U^{-1}FDF^{-1}U=(\mathrm {URF,UFL,BDR} )$,

the so called "Z move? Now the three pieces are no longer in the same plane and there is no obvious "correct" orientation. In the first case, I can, in addition, apply standard twisting algorithms and it is obvious I have done so because the U orientation is broken, but in the second case it is not obvious at all. Furthermore, there are groups such as,

$D^{2}R^{-1}DRU^{-1}R^{-1}D^{-1}RUD^{2}=(\mathrm {URF,UFL,RBD} )$

that achieve the same result, but do not explicitly employ any twisting algorithms.

I believe that there is not a unique untwisted orientation and that one must arbitrarily choose an orientation for any particular group and consider all other groups of the same pieces to be "twisted". The author seems to admit as much, "[f]or this subgroup there are more choices, depending on the precise way you fix the orientation" and later "[f]or technical reasons, the above analysis is not complete", without ever really explaining what the problem is.

A much cleaner way of stating this, in my opinion, is "permutations of the piece positions ignoring differing orientations". It appears only to have been stated in the way that it is to emphasise an isomorphism with the C_o group. SpinningSpark 15:20, 30 December 2014 (UTC)

Thanks—I think you're right, there are multiple ways to define orientation. After some thinking and consulting Douglas Hofstadter's Metamagical Themas, I've written out one definition that I believe is consistent with the definition of $C_{p}$ given in the article, and added it in a footnote. (I've tried to phrase it comprehensibly, but improvements in clarity would be welcome.)

As for your point that $C_{p}$ can also be thought of as "permutations of the piece positions ignoring differing orientations"—I think that's correct. More formally, $C_{p}$ is isomorphic to the quotient group of $G$ with respect to $C_{o}$—a consequence of the fact that $G=C_{o}\rtimes C_{p}$. —Granger (talk · contribs) 19:36, 30 December 2014 (UTC)

...and while we are on the subject...

"For technical reasons, the above analysis is not complete. However, the possible permutations of the cubes, even when ignoring the orientations of the said cubes, is at the same time no bigger than Cp and at least as big as Cp, and this means that the cube group is the semi-direct product given above."

Does that even mean anything? "No bigger than Cp and at least as big as Cp" simply means equal to Cp doesn't it? And Cp is the permutations ignoring orientation so why is it necessary to explicitly qualify that? SpinningSpark 15:52, 30 December 2014 (UTC)

Yeah. I can't tell what this parenthetical note is trying to say either. —Granger (talk · contribs) 19:37, 30 December 2014 (UTC)

One to One Correspondence?

I am pretty sure that there is not a one to one correspondense between sets of moves and positions.

Take the example of no moves and RUR'U' six times. Since it repeats the starting position after all six repititions, it will correspond to the same position as no moves, making it not a one to one correspondense (since there exists a position with more than one set of moves). An Autist (talk) 19:07, 11 October 2018 (UTC)

It's not clear what you mean by the "set of moves". If you mean the various strings that can represent sequences of moves, then no. But that's not what the article says: "Indeed with the solved position as a starting point, there is a one-to-one correspondence between each of the legal positions of the Rubik's Cube and the elements of G." And that much is certainly fine. –Deacon Vorbis (carbon • videos) 19:13, 11 October 2018 (UTC)

Of course there isn't. There's an infinite number of notation strings and a finite number of positions. Any move sequence repeated often enough will return to the solved position.

The use of the phrase "cube move" may be a bit confusing here. It does not describe the actual moves, but what those moves do. Think of it like PLLs. A T-perm is a T-perm, no matter which algorithm you use. "R U R' U' R' F R2 U' R' U' R U' R' F'" is just one representation the T-perm. Why is called "cube move"? Most likely because you can add cube moves on top of each other, no matter which version you use. A T-perm plus an H-perm will always result in an F-perm, no matter which version of those you use.

"Each element of the set G corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces." - This is just how the group is defined. The elements of the group aren't the scramble sequences or alg notations, but the result of what those sequences do.

You could create a group where all different scramble sequences are different group elements. But that would be very boring and not helpful for the Rubik's cube. It would be just words with a 6-letter-alphabet. Judith Sunrise (talk) 23:02, 11 October 2018 (UTC)

A group element is different from the ways that it can be composed out of other group elements. For example, LL' and RR' are the same group element (the identity element). It's just that it can be composed in many different ways. In the (abelian) group of natural numbers for example, the number 7 can be composed from two elements as 0+7, 1+6, 2+5, or 3+4, but the four ways obviously yield the same number. Esquivalience (talk) 01:57, 12 October 2018 (UTC)

Moves v move contradiction.

"Each element of the set G {\displaystyle G} G corresponds to a cube move, which is the effect of any sequence of rotations of the cube's faces. With this representation, not only can any cube move be represented, but also any position of the cube as well, by detailing the cube moves required to rotate the solved cube into that position." seems to contain a contradiction or at least inconsistency because surely it should say "cube move required" to be consistent with the definition of a cube move earlier in the paragraph? If a cube move is the effect of any *sequence of rotations of the cube's faces* detailing a single cube move would be enough to represent any position of the cube. Or am I mistaken? Arctic Gazelle (talk) 21:02, 11 July 2021 (UTC)

I suppose the distinction is that the move required to reach a position is not unique. A position can be specified by specifying the move to get there, but it is not the only possible move. Usually, there will be a large number of moves available for any desired result. What is not correct is that any position of the cube can be represented by a move. Only one twelth of the possible positions of the pieces can be reached by a cube move. A fact you will discover if you try to disassemble and randomly reassemble the cube pieces. It's only the members of G that can be reached. That's even worse for S₄₈. Only a very small fraction of that group can be reached. One has to unpeel and reattach the stickers to get to the rest SpinningSpark 14:19, 15 August 2021 (UTC)

term

The article says facet a lot, isn't it meant to say facelet? — Preceding unsigned comment added by 85.218.239.1 (talk) 02:15, 31 March 2022 (UTC)

Notation confusion

In section 1 "Cube Moves", the notation uses F, U, B, etc, for clockwise moves, and uses the apostrophe ' to indicate counter clockwise moves. The superscript ^2 indicates a rotation of 180 degrees. All good.

In section 2, "Group Structure", the superscript ^{-1} is used. This notation is not introduced anywhere. I presume that ^{-1} means counter clockwise? If so, would it be appropriate to replace the ^{-1} with an apostrophe for consistency? If not, can we have a note about the meaning of ^{-1}? Call me Bifficus (talk) 14:35, 28 September 2023 (UTC)